| L(s) = 1 | + 3-s + 2·5-s − 8·7-s + 9-s + 8·11-s + 2·15-s − 12·17-s − 8·21-s − 25-s + 27-s + 8·33-s − 16·35-s + 8·43-s + 2·45-s + 34·49-s − 12·51-s + 20·53-s + 16·55-s − 8·59-s + 12·61-s − 8·63-s + 8·67-s − 32·71-s − 75-s − 64·77-s + 81-s − 24·85-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 3.02·7-s + 1/3·9-s + 2.41·11-s + 0.516·15-s − 2.91·17-s − 1.74·21-s − 1/5·25-s + 0.192·27-s + 1.39·33-s − 2.70·35-s + 1.21·43-s + 0.298·45-s + 34/7·49-s − 1.68·51-s + 2.74·53-s + 2.15·55-s − 1.04·59-s + 1.53·61-s − 1.00·63-s + 0.977·67-s − 3.79·71-s − 0.115·75-s − 7.29·77-s + 1/9·81-s − 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 691200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 691200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.483266537012718160246988734430, −7.28106910234984466129499703268, −6.91469103626909406223255043804, −6.89223743244911036927037618218, −6.21235484024916055277814682123, −6.15267013015603292112906972931, −5.71236013705473716417431154824, −4.58601025014912673368663448129, −4.10625798014363516984369255368, −3.81183966821478542109453108564, −3.30547293943726997892238481319, −2.44821729992400328434125228202, −2.33250140292139670172265163731, −1.21505631432587485936766209289, 0,
1.21505631432587485936766209289, 2.33250140292139670172265163731, 2.44821729992400328434125228202, 3.30547293943726997892238481319, 3.81183966821478542109453108564, 4.10625798014363516984369255368, 4.58601025014912673368663448129, 5.71236013705473716417431154824, 6.15267013015603292112906972931, 6.21235484024916055277814682123, 6.89223743244911036927037618218, 6.91469103626909406223255043804, 7.28106910234984466129499703268, 8.483266537012718160246988734430
